Abstract

This paper contains several problems that can be formulated
mathematically as two-dimensional boundary value problems for partial differential equations containing a parameter. A method is given which leads directly to asymptotic solutions for large values of the parameter without resorting to the exact solutions. The examples discussed involve linear differential equations and are drawn primarily from various problems in the theory of elasticity.
The method involves consideration of what are termed corner-layers in addition to the well known boundary-layers. The need for considering these corner-layers arises from the fact that the problems treated lead to boundary-layer differential equations which contain derivatives, not only with respect to the boundary-layer variable, but also with respect to the remaining independent variable. Thus, the
solution of such boundary-layer equations requires knowledge of boundary conditions in addition to those needed in standard boundary-layer problems.
The applications include: a heat conduction problem, two problems with transverse bending of stretched plates, and two problems from elastic shell theory.
The shell problems concern the bending of both the shallow and the non-shallow helicoidal shell. It is found that these shells have boundary-layers whose characteristic length is proportional to the one-third power of the thickness parameter. This may be contrasted with shells
of revolution, where this characteristic length is proportional to the one-half power of the thickness parameter.